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<div class="section">
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="boost_numericconversion.definitions"></a><a class="link" href="definitions.html" title="Definitions">Definitions</a>
</h2></div></div></div>
<div class="toc"><dl class="toc">
<dt><span class="section"><a href="definitions.html#boost_numericconversion.definitions.introduction">Introduction</a></span></dt>
<dt><span class="section"><a href="definitions.html#boost_numericconversion.definitions.types_and_values">Types
      and Values</a></span></dt>
<dt><span class="section"><a href="definitions.html#boost_numericconversion.definitions.c___arithmetic_types">C++
      Arithmetic Types</a></span></dt>
<dt><span class="section"><a href="definitions.html#boost_numericconversion.definitions.numeric_types">Numeric
      Types</a></span></dt>
<dt><span class="section"><a href="definitions.html#boost_numericconversion.definitions.range_and_precision">Range
      and Precision</a></span></dt>
<dt><span class="section"><a href="definitions.html#boost_numericconversion.definitions.exact__correctly_rounded_and_out_of_range_representations">Exact,
      Correctly Rounded and Out-Of-Range Representations</a></span></dt>
<dt><span class="section"><a href="definitions.html#boost_numericconversion.definitions.standard__numeric__conversions">Standard
      (numeric) Conversions</a></span></dt>
<dt><span class="section"><a href="definitions.html#boost_numericconversion.definitions.subranged_conversion_direction__subtype_and_supertype">Subranged
      Conversion Direction, Subtype and Supertype</a></span></dt>
</dl></div>
<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="boost_numericconversion.definitions.introduction"></a><a class="link" href="definitions.html#boost_numericconversion.definitions.introduction" title="Introduction">Introduction</a>
</h3></div></div></div>
<p>
        This section provides definitions of terms used in the Numeric Conversion
        library.
      </p>
<div class="blurb">
<div class="titlepage"><div><div><p class="title"><b></b></p></div></div></div>
<p>
        <span class="bold"><strong>Notation</strong></span> <span class="underline">underlined
        text</span> denotes terms defined in the C++ standard.
      </p>
<p>
        <span class="bold"><strong>bold face</strong></span> denotes terms defined here but
        not in the standard.
      </p>
</div>
</div>
<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="boost_numericconversion.definitions.types_and_values"></a><a class="link" href="definitions.html#boost_numericconversion.definitions.types_and_values" title="Types and Values">Types
      and Values</a>
</h3></div></div></div>
<p>
        As defined by the <span class="underline">C++ Object Model</span>
        (§1.7) the <span class="underline">storage</span> or memory on which
        a C++ program runs is a contiguous sequence of <span class="underline">bytes</span>
        where each byte is a contiguous sequence of bits.
      </p>
<p>
        An <span class="underline">object</span> is a region of storage (§1.8)
        and has a type (§3.9).
      </p>
<p>
        A <span class="underline">type</span> is a discrete set of values.
      </p>
<p>
        An object of type <code class="computeroutput"><span class="identifier">T</span></code> has an
        <span class="underline">object representation</span> which is the
        sequence of bytes stored in the object (§3.9/4)
      </p>
<p>
        An object of type <code class="computeroutput"><span class="identifier">T</span></code> has a
        <span class="underline">value representation</span> which is the set
        of bits that determine the <span class="emphasis"><em>value</em></span> of an object of that
        type (§3.9/4). For <span class="underline">POD</span> types (§3.9/10),
        this bitset is given by the object representation, but not all the bits in
        the storage need to participate in the value representation (except for character
        types): for example, some bits might be used for padding or there may be
        trap-bits.
      </p>
<p>
        <span class="inlinemediaobject"><img src="../images/space.png" alt="space"></span>
      </p>
<p>
        The <span class="bold"><strong>typed value</strong></span> that is held by an object
        is the value which is determined by its value representation.
      </p>
<p>
        An <span class="bold"><strong>abstract value</strong></span> (untyped) is the conceptual
        information that is represented in a type (i.e. the number π).
      </p>
<p>
        The <span class="bold"><strong>intrinsic value</strong></span> of an object is the
        binary value of the sequence of unsigned characters which form its object
        representation.
      </p>
<p>
        <span class="inlinemediaobject"><img src="../images/space.png" alt="space"></span>
      </p>
<p>
        <span class="emphasis"><em>Abstract</em></span> values can be <span class="bold"><strong>represented</strong></span>
        in a given type.
      </p>
<p>
        To <span class="bold"><strong>represent</strong></span> an abstract value <code class="computeroutput"><span class="identifier">V</span></code> in a type <code class="computeroutput"><span class="identifier">T</span></code>
        is to obtain a typed value <code class="computeroutput"><span class="identifier">v</span></code>
        which corresponds to the abstract value <code class="computeroutput"><span class="identifier">V</span></code>.
      </p>
<p>
        The operation is denoted using the <code class="computeroutput"><span class="identifier">rep</span><span class="special">()</span></code> operator, as in: <code class="computeroutput"><span class="identifier">v</span><span class="special">=</span><span class="identifier">rep</span><span class="special">(</span><span class="identifier">V</span><span class="special">)</span></code>. <code class="computeroutput"><span class="identifier">v</span></code> is the <span class="bold"><strong>representation</strong></span>
        of <code class="computeroutput"><span class="identifier">V</span></code> in the type <code class="computeroutput"><span class="identifier">T</span></code>.
      </p>
<p>
        For example, the abstract value π can be represented in the type <code class="computeroutput"><span class="keyword">double</span></code> as the <code class="computeroutput"><span class="keyword">double</span>
        <span class="identifier">value</span> <span class="identifier">M_PI</span></code>
        and in the type <code class="computeroutput"><span class="keyword">int</span></code> as the
        <code class="computeroutput"><span class="keyword">int</span> <span class="identifier">value</span>
        <span class="number">3</span></code>
      </p>
<p>
        <span class="inlinemediaobject"><img src="../images/space.png" alt="space"></span>
      </p>
<p>
        Conversely, <span class="emphasis"><em>typed values</em></span> can be <span class="bold"><strong>abstracted</strong></span>.
      </p>
<p>
        To <span class="bold"><strong>abstract</strong></span> a typed value <code class="computeroutput"><span class="identifier">v</span></code> of type <code class="computeroutput"><span class="identifier">T</span></code>
        is to obtain the abstract value <code class="computeroutput"><span class="identifier">V</span></code>
        whose representation in <code class="computeroutput"><span class="identifier">T</span></code>
        is <code class="computeroutput"><span class="identifier">v</span></code>.
      </p>
<p>
        The operation is denoted using the <code class="computeroutput"><span class="identifier">abt</span><span class="special">()</span></code> operator, as in: <code class="computeroutput"><span class="identifier">V</span><span class="special">=</span><span class="identifier">abt</span><span class="special">(</span><span class="identifier">v</span><span class="special">)</span></code>.
      </p>
<p>
        <code class="computeroutput"><span class="identifier">V</span></code> is the <span class="bold"><strong>abstraction</strong></span>
        of <code class="computeroutput"><span class="identifier">v</span></code> of type <code class="computeroutput"><span class="identifier">T</span></code>.
      </p>
<p>
        Abstraction is just an abstract operation (you can't do it); but it is defined
        nevertheless because it will be used to give the definitions in the rest
        of this document.
      </p>
</div>
<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="boost_numericconversion.definitions.c___arithmetic_types"></a><a class="link" href="definitions.html#boost_numericconversion.definitions.c___arithmetic_types" title="C++ Arithmetic Types">C++
      Arithmetic Types</a>
</h3></div></div></div>
<p>
        The C++ language defines <span class="underline">fundamental types</span>
        (§3.9.1). The following subsets of the fundamental types are intended to
        represent <span class="emphasis"><em>numbers</em></span>:
      </p>
<div class="variablelist">
<p class="title"><b></b></p>
<dl class="variablelist">
<dt><span class="term"><span class="underline">signed integer types</span> (§3.9.1/2):</span></dt>
<dd><p>
              <code class="computeroutput"><span class="special">{</span><span class="keyword">signed</span>
              <span class="keyword">char</span><span class="special">,</span>
              <span class="keyword">signed</span> <span class="keyword">short</span>
              <span class="keyword">int</span><span class="special">,</span>
              <span class="keyword">signed</span> <span class="keyword">int</span><span class="special">,</span> <span class="keyword">signed</span> <span class="keyword">long</span> <span class="keyword">int</span><span class="special">}</span></code> Can be used to represent general integer
              numbers (both negative and positive).
            </p></dd>
<dt><span class="term"><span class="underline">unsigned integer types</span> (§3.9.1/3):</span></dt>
<dd><p>
              <code class="computeroutput"><span class="special">{</span><span class="keyword">unsigned</span>
              <span class="keyword">char</span><span class="special">,</span>
              <span class="keyword">unsigned</span> <span class="keyword">short</span>
              <span class="keyword">int</span><span class="special">,</span>
              <span class="keyword">unsigned</span> <span class="keyword">int</span><span class="special">,</span> <span class="keyword">unsigned</span>
              <span class="keyword">long</span> <span class="keyword">int</span><span class="special">}</span></code> Can be used to represent positive
              integer numbers with modulo-arithmetic.
            </p></dd>
<dt><span class="term"><span class="underline">floating-point types</span> (§3.9.1/8):</span></dt>
<dd><p>
              <code class="computeroutput"><span class="special">{</span><span class="keyword">float</span><span class="special">,</span><span class="keyword">double</span><span class="special">,</span><span class="keyword">long</span> <span class="keyword">double</span><span class="special">}</span></code>
              Can be used to represent real numbers.
            </p></dd>
<dt><span class="term"><span class="underline">integral or integer types</span> (§3.9.1/7):</span></dt>
<dd><p>
              <code class="computeroutput"><span class="special">{{</span><span class="keyword">signed</span>
              <span class="identifier">integers</span><span class="special">},{</span><span class="keyword">unsigned</span> <span class="identifier">integers</span><span class="special">},</span> <span class="keyword">bool</span><span class="special">,</span> <span class="keyword">char</span> <span class="keyword">and</span> <span class="keyword">wchar_t</span><span class="special">}</span></code>
            </p></dd>
<dt><span class="term"><span class="underline">arithmetic types</span> (§3.9.1/8):</span></dt>
<dd><p>
              <code class="computeroutput"><span class="special">{{</span><span class="identifier">integer</span>
              <span class="identifier">types</span><span class="special">},{</span><span class="identifier">floating</span> <span class="identifier">types</span><span class="special">}}</span></code>
            </p></dd>
</dl>
</div>
<p>
        The integer types are required to have a <span class="emphasis"><em>binary</em></span> value
        representation.
      </p>
<p>
        Additionally, the signed/unsigned integer types of the same base type (<code class="computeroutput"><span class="keyword">short</span></code>, <code class="computeroutput"><span class="keyword">int</span></code>
        or <code class="computeroutput"><span class="keyword">long</span></code>) are required to have
        the same value representation, that is:
      </p>
<pre class="programlisting">         <span class="keyword">int</span> <span class="identifier">i</span> <span class="special">=</span> <span class="special">-</span><span class="number">3</span> <span class="special">;</span> <span class="comment">// suppose value representation is: 10011 (sign bit + 4 magnitude bits)</span>
<span class="keyword">unsigned</span> <span class="keyword">int</span> <span class="identifier">u</span> <span class="special">=</span>  <span class="identifier">i</span> <span class="special">;</span> <span class="comment">// u is required to have the same 10011 as its value representation.</span>
</pre>
<p>
        In other words, the integer types signed/unsigned X use the same value representation
        but a different <span class="emphasis"><em>interpretation</em></span> of it; that is, their
        <span class="emphasis"><em>typed values</em></span> might differ.
      </p>
<p>
        Another consequence of this is that the range for signed X is always a smaller
        subset of the range of unsigned X, as required by §3.9.1/3.
      </p>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top">
<p>
          Always remember that unsigned types, unlike signed types, have modulo-arithmetic;
          that is, they do not overflow. This means that:
        </p>
<p>
          <span class="bold"><strong>-</strong></span> Always be extra careful when mixing
          signed/unsigned types
        </p>
<p>
          <span class="bold"><strong>-</strong></span> Use unsigned types only when you need
          modulo arithmetic or very very large numbers. Don't use unsigned types
          just because you intend to deal with positive values only (you can do this
          with signed types as well).
        </p>
</td></tr>
</table></div>
</div>
<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="boost_numericconversion.definitions.numeric_types"></a><a class="link" href="definitions.html#boost_numericconversion.definitions.numeric_types" title="Numeric Types">Numeric
      Types</a>
</h3></div></div></div>
<p>
        This section introduces the following definitions intended to integrate arithmetic
        types with user-defined types which behave like numbers. Some definitions
        are purposely broad in order to include a vast variety of user-defined number
        types.
      </p>
<p>
        Within this library, the term <span class="emphasis"><em>number</em></span> refers to an abstract
        numeric value.
      </p>
<p>
        A type is <span class="bold"><strong>numeric</strong></span> if:
      </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
            It is an arithmetic type, or,
          </li>
<li class="listitem">
            It is a user-defined type which
            <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: circle; ">
<li class="listitem">
                  Represents numeric abstract values (i.e. numbers).
                </li>
<li class="listitem">
                  Can be converted (either implicitly or explicitly) to/from at least
                  one arithmetic type.
                </li>
<li class="listitem">
                  Has <a class="link" href="definitions.html#boost_numericconversion.definitions.range_and_precision" title="Range and Precision">range</a>
                  (possibly unbounded) and <a class="link" href="definitions.html#boost_numericconversion.definitions.range_and_precision" title="Range and Precision">precision</a>
                  (possibly dynamic or unlimited).
                </li>
<li class="listitem">
                  Provides an specialization of <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span></code>.
                </li>
</ul></div>
          </li>
</ul></div>
<p>
        A numeric type is <span class="bold"><strong>signed</strong></span> if the abstract
        values it represent include negative numbers.
      </p>
<p>
        A numeric type is <span class="bold"><strong>unsigned</strong></span> if the abstract
        values it represent exclude negative numbers.
      </p>
<p>
        A numeric type is <span class="bold"><strong>modulo</strong></span> if it has modulo-arithmetic
        (does not overflow).
      </p>
<p>
        A numeric type is <span class="bold"><strong>integer</strong></span> if the abstract
        values it represent are whole numbers.
      </p>
<p>
        A numeric type is <span class="bold"><strong>floating</strong></span> if the abstract
        values it represent are real numbers.
      </p>
<p>
        An <span class="bold"><strong>arithmetic value</strong></span> is the typed value of
        an arithmetic type
      </p>
<p>
        A <span class="bold"><strong>numeric value</strong></span> is the typed value of a
        numeric type
      </p>
<p>
        These definitions simply generalize the standard notions of arithmetic types
        and values by introducing a superset called <span class="underline">numeric</span>.
        All arithmetic types and values are numeric types and values, but not vice
        versa, since user-defined numeric types are not arithmetic types.
      </p>
<p>
        The following examples clarify the differences between arithmetic and numeric
        types (and values):
      </p>
<pre class="programlisting"><span class="comment">// A numeric type which is not an arithmetic type (is user-defined)</span>
<span class="comment">// and which is intended to represent integer numbers (i.e., an 'integer' numeric type)</span>
<span class="keyword">class</span> <span class="identifier">MyInt</span>
<span class="special">{</span>
    <span class="identifier">MyInt</span> <span class="special">(</span> <span class="keyword">long</span> <span class="keyword">long</span> <span class="identifier">v</span> <span class="special">)</span> <span class="special">;</span>
    <span class="keyword">long</span> <span class="keyword">long</span> <span class="identifier">to_builtin</span><span class="special">();</span>
<span class="special">}</span> <span class="special">;</span>
<span class="keyword">namespace</span> <span class="identifier">std</span> <span class="special">{</span>
<span class="keyword">template</span><span class="special">&lt;&gt;</span> <span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">MyInt</span><span class="special">&gt;</span> <span class="special">{</span> <span class="special">...</span> <span class="special">}</span> <span class="special">;</span>
<span class="special">}</span>

<span class="comment">// A 'floating' numeric type (double) which is also an arithmetic type (built-in),</span>
<span class="comment">// with a float numeric value.</span>
<span class="keyword">double</span> <span class="identifier">pi</span> <span class="special">=</span> <span class="identifier">M_PI</span> <span class="special">;</span>

<span class="comment">// A 'floating' numeric type with a whole numeric value.</span>
<span class="comment">// NOTE: numeric values are typed valued, hence, they are, for instance,</span>
<span class="comment">// integer or floating, despite the value itself being whole or including</span>
<span class="comment">// a fractional part.</span>
<span class="keyword">double</span> <span class="identifier">two</span> <span class="special">=</span> <span class="number">2.0</span> <span class="special">;</span>

<span class="comment">// An integer numeric type with an integer numeric value.</span>
<span class="identifier">MyInt</span> <span class="identifier">i</span><span class="special">(</span><span class="number">1234</span><span class="special">);</span>
</pre>
</div>
<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="boost_numericconversion.definitions.range_and_precision"></a><a class="link" href="definitions.html#boost_numericconversion.definitions.range_and_precision" title="Range and Precision">Range
      and Precision</a>
</h3></div></div></div>
<p>
        Given a number set <code class="computeroutput"><span class="identifier">N</span></code>, some
        of its elements are representable in a numeric type <code class="computeroutput"><span class="identifier">T</span></code>.
      </p>
<p>
        The set of representable values of type <code class="computeroutput"><span class="identifier">T</span></code>,
        or numeric set of <code class="computeroutput"><span class="identifier">T</span></code>, is a
        set of numeric values whose elements are the representation of some subset
        of <code class="computeroutput"><span class="identifier">N</span></code>.
      </p>
<p>
        For example, the interval of <code class="computeroutput"><span class="keyword">int</span></code>
        values <code class="computeroutput"><span class="special">[</span><span class="identifier">INT_MIN</span><span class="special">,</span><span class="identifier">INT_MAX</span><span class="special">]</span></code> is the set of representable values of type
        <code class="computeroutput"><span class="keyword">int</span></code>, i.e. the <code class="computeroutput"><span class="keyword">int</span></code> numeric set, and corresponds to the representation
        of the elements of the interval of abstract values <code class="computeroutput"><span class="special">[</span><span class="identifier">abt</span><span class="special">(</span><span class="identifier">INT_MIN</span><span class="special">),</span><span class="identifier">abt</span><span class="special">(</span><span class="identifier">INT_MAX</span><span class="special">)]</span></code>
        from the integer numbers.
      </p>
<p>
        Similarly, the interval of <code class="computeroutput"><span class="keyword">double</span></code>
        values <code class="computeroutput"><span class="special">[-</span><span class="identifier">DBL_MAX</span><span class="special">,</span><span class="identifier">DBL_MAX</span><span class="special">]</span></code> is the <code class="computeroutput"><span class="keyword">double</span></code>
        numeric set, which corresponds to the subset of the real numbers from <code class="computeroutput"><span class="identifier">abt</span><span class="special">(-</span><span class="identifier">DBL_MAX</span><span class="special">)</span></code> to <code class="computeroutput"><span class="identifier">abt</span><span class="special">(</span><span class="identifier">DBL_MAX</span><span class="special">)</span></code>.
      </p>
<p>
        <span class="inlinemediaobject"><img src="../images/space.png" alt="space"></span>
      </p>
<p>
        Let <span class="bold"><strong><code class="computeroutput"><span class="identifier">next</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span></code></strong></span>
        denote the lowest numeric value greater than x.
      </p>
<p>
        Let <span class="bold"><strong><code class="computeroutput"><span class="identifier">prev</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span></code></strong></span>
        denote the highest numeric value lower then x.
      </p>
<p>
        Let <span class="bold"><strong><code class="computeroutput"><span class="identifier">v</span><span class="special">=</span><span class="identifier">prev</span><span class="special">(</span><span class="identifier">next</span><span class="special">(</span><span class="identifier">V</span><span class="special">))</span></code></strong></span> and <span class="bold"><strong><code class="computeroutput"><span class="identifier">v</span><span class="special">=</span><span class="identifier">next</span><span class="special">(</span><span class="identifier">prev</span><span class="special">(</span><span class="identifier">V</span><span class="special">))</span></code></strong></span>
        be identities that relate a numeric typed value <code class="computeroutput"><span class="identifier">v</span></code>
        with a number <code class="computeroutput"><span class="identifier">V</span></code>.
      </p>
<p>
        An ordered pair of numeric values <code class="computeroutput"><span class="identifier">x</span></code>,<code class="computeroutput"><span class="identifier">y</span></code> s.t. <code class="computeroutput"><span class="identifier">x</span><span class="special">&lt;</span><span class="identifier">y</span></code> are
        <span class="bold"><strong>consecutive</strong></span> iff <code class="computeroutput"><span class="identifier">next</span><span class="special">(</span><span class="identifier">x</span><span class="special">)==</span><span class="identifier">y</span></code>.
      </p>
<p>
        The abstract distance between consecutive numeric values is usually referred
        to as a <span class="underline">Unit in the Last Place</span>, or
        <span class="bold"><strong>ulp</strong></span> for short. A ulp is a quantity whose
        abstract magnitude is relative to the numeric values it corresponds to: If
        the numeric set is not evenly distributed, that is, if the abstract distance
        between consecutive numeric values varies along the set -as is the case with
        the floating-point types-, the magnitude of 1ulp after the numeric value
        <code class="computeroutput"><span class="identifier">x</span></code> might be (usually is) different
        from the magnitude of a 1ulp after the numeric value y for <code class="computeroutput"><span class="identifier">x</span><span class="special">!=</span><span class="identifier">y</span></code>.
      </p>
<p>
        Since numbers are inherently ordered, a <span class="bold"><strong>numeric set</strong></span>
        of type <code class="computeroutput"><span class="identifier">T</span></code> is an ordered sequence
        of numeric values (of type <code class="computeroutput"><span class="identifier">T</span></code>)
        of the form:
      </p>
<pre class="programlisting"><span class="identifier">REP</span><span class="special">(</span><span class="identifier">T</span><span class="special">)={</span><span class="identifier">l</span><span class="special">,</span><span class="identifier">next</span><span class="special">(</span><span class="identifier">l</span><span class="special">),</span><span class="identifier">next</span><span class="special">(</span><span class="identifier">next</span><span class="special">(</span><span class="identifier">l</span><span class="special">)),...,</span><span class="identifier">prev</span><span class="special">(</span><span class="identifier">prev</span><span class="special">(</span><span class="identifier">h</span><span class="special">)),</span><span class="identifier">prev</span><span class="special">(</span><span class="identifier">h</span><span class="special">),</span><span class="identifier">h</span><span class="special">}</span>
</pre>
<p>
        where <code class="computeroutput"><span class="identifier">l</span></code> and <code class="computeroutput"><span class="identifier">h</span></code> are respectively the lowest and highest
        values of type <code class="computeroutput"><span class="identifier">T</span></code>, called
        the boundary values of type <code class="computeroutput"><span class="identifier">T</span></code>.
      </p>
<p>
        <span class="inlinemediaobject"><img src="../images/space.png" alt="space"></span>
      </p>
<p>
        A numeric set is discrete. It has a <span class="bold"><strong>size</strong></span>
        which is the number of numeric values in the set, a <span class="bold"><strong>width</strong></span>
        which is the abstract difference between the highest and lowest boundary
        values: <code class="computeroutput"><span class="special">[</span><span class="identifier">abt</span><span class="special">(</span><span class="identifier">h</span><span class="special">)-</span><span class="identifier">abt</span><span class="special">(</span><span class="identifier">l</span><span class="special">)]</span></code>, and a <span class="bold"><strong>density</strong></span>
        which is the relation between its size and width: <code class="computeroutput"><span class="identifier">density</span><span class="special">=</span><span class="identifier">size</span><span class="special">/</span><span class="identifier">width</span></code>.
      </p>
<p>
        The integer types have density 1, which means that there are no unrepresentable
        integer numbers between <code class="computeroutput"><span class="identifier">abt</span><span class="special">(</span><span class="identifier">l</span><span class="special">)</span></code>
        and <code class="computeroutput"><span class="identifier">abt</span><span class="special">(</span><span class="identifier">h</span><span class="special">)</span></code> (i.e.
        there are no gaps). On the other hand, floating types have density much smaller
        than 1, which means that there are real numbers unrepresented between consecutive
        floating values (i.e. there are gaps).
      </p>
<p>
        <span class="inlinemediaobject"><img src="../images/space.png" alt="space"></span>
      </p>
<p>
        The interval of <span class="underline">abstract values</span> <code class="computeroutput"><span class="special">[</span><span class="identifier">abt</span><span class="special">(</span><span class="identifier">l</span><span class="special">),</span><span class="identifier">abt</span><span class="special">(</span><span class="identifier">h</span><span class="special">)]</span></code>
        is the range of the type <code class="computeroutput"><span class="identifier">T</span></code>,
        denoted <code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="identifier">T</span><span class="special">)</span></code>.
      </p>
<p>
        A range is a set of abstract values and not a set of numeric values. In other
        documents, such as the C++ standard, the word <code class="computeroutput"><span class="identifier">range</span></code>
        is <span class="emphasis"><em>sometimes</em></span> used as synonym for <code class="computeroutput"><span class="identifier">numeric</span>
        <span class="identifier">set</span></code>, that is, as the ordered sequence
        of numeric values from <code class="computeroutput"><span class="identifier">l</span></code>
        to <code class="computeroutput"><span class="identifier">h</span></code>. In this document, however,
        a range is an abstract interval which subtends the numeric set.
      </p>
<p>
        For example, the sequence <code class="computeroutput"><span class="special">[-</span><span class="identifier">DBL_MAX</span><span class="special">,</span><span class="identifier">DBL_MAX</span><span class="special">]</span></code>
        is the numeric set of the type <code class="computeroutput"><span class="keyword">double</span></code>,
        and the real interval <code class="computeroutput"><span class="special">[</span><span class="identifier">abt</span><span class="special">(-</span><span class="identifier">DBL_MAX</span><span class="special">),</span><span class="identifier">abt</span><span class="special">(</span><span class="identifier">DBL_MAX</span><span class="special">)]</span></code>
        is its range.
      </p>
<p>
        Notice, for instance, that the range of a floating-point type is <span class="emphasis"><em>continuous</em></span>
        unlike its numeric set.
      </p>
<p>
        This definition was chosen because:
      </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
            <span class="bold"><strong>(a)</strong></span> The discrete set of numeric values
            is already given by the numeric set.
          </li>
<li class="listitem">
            <span class="bold"><strong>(b)</strong></span> Abstract intervals are easier to
            compare and overlap since only boundary values need to be considered.
          </li>
</ul></div>
<p>
        This definition allows for a concise definition of <code class="computeroutput"><span class="identifier">subranged</span></code>
        as given in the last section.
      </p>
<p>
        The width of a numeric set, as defined, is exactly equivalent to the width
        of a range.
      </p>
<p>
        <span class="inlinemediaobject"><img src="../images/space.png" alt="space"></span>
      </p>
<p>
        The <span class="bold"><strong>precision</strong></span> of a type is given by the
        width or density of the numeric set.
      </p>
<p>
        For integer types, which have density 1, the precision is conceptually equivalent
        to the range and is determined by the number of bits used in the value representation:
        The higher the number of bits the bigger the size of the numeric set, the
        wider the range, and the higher the precision.
      </p>
<p>
        For floating types, which have density &lt;&lt;1, the precision is given
        not by the width of the range but by the density. In a typical implementation,
        the range is determined by the number of bits used in the exponent, and the
        precision by the number of bits used in the mantissa (giving the maximum
        number of significant digits that can be exactly represented). The higher
        the number of exponent bits the wider the range, while the higher the number
        of mantissa bits, the higher the precision.
      </p>
</div>
<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="boost_numericconversion.definitions.exact__correctly_rounded_and_out_of_range_representations"></a><a class="link" href="definitions.html#boost_numericconversion.definitions.exact__correctly_rounded_and_out_of_range_representations" title="Exact, Correctly Rounded and Out-Of-Range Representations">Exact,
      Correctly Rounded and Out-Of-Range Representations</a>
</h3></div></div></div>
<p>
        Given an abstract value <code class="computeroutput"><span class="identifier">V</span></code>
        and a type <code class="computeroutput"><span class="identifier">T</span></code> with its corresponding
        range <code class="computeroutput"><span class="special">[</span><span class="identifier">abt</span><span class="special">(</span><span class="identifier">l</span><span class="special">),</span><span class="identifier">abt</span><span class="special">(</span><span class="identifier">h</span><span class="special">)]</span></code>:
      </p>
<p>
        If <code class="computeroutput"><span class="identifier">V</span> <span class="special">&lt;</span>
        <span class="identifier">abt</span><span class="special">(</span><span class="identifier">l</span><span class="special">)</span></code> or
        <code class="computeroutput"><span class="identifier">V</span> <span class="special">&gt;</span>
        <span class="identifier">abt</span><span class="special">(</span><span class="identifier">h</span><span class="special">)</span></code>, <code class="computeroutput"><span class="identifier">V</span></code> is <span class="bold"><strong>not representable</strong></span>
        (cannot be represented) in the type <code class="computeroutput"><span class="identifier">T</span></code>,
        or, equivalently, it's representation in the type <code class="computeroutput"><span class="identifier">T</span></code>
        is <span class="bold"><strong>out of range</strong></span>, or <span class="bold"><strong>overflows</strong></span>.
      </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
            If <code class="computeroutput"><span class="identifier">V</span> <span class="special">&lt;</span>
            <span class="identifier">abt</span><span class="special">(</span><span class="identifier">l</span><span class="special">)</span></code>,
            the <span class="bold"><strong>overflow is negative</strong></span>.
          </li>
<li class="listitem">
            If <code class="computeroutput"><span class="identifier">V</span> <span class="special">&gt;</span>
            <span class="identifier">abt</span><span class="special">(</span><span class="identifier">h</span><span class="special">)</span></code>,
            the <span class="bold"><strong>overflow is positive</strong></span>.
          </li>
</ul></div>
<p>
        If <code class="computeroutput"><span class="identifier">V</span> <span class="special">&gt;=</span>
        <span class="identifier">abt</span><span class="special">(</span><span class="identifier">l</span><span class="special">)</span></code> and
        <code class="computeroutput"><span class="identifier">V</span> <span class="special">&lt;=</span>
        <span class="identifier">abt</span><span class="special">(</span><span class="identifier">h</span><span class="special">)</span></code>, <code class="computeroutput"><span class="identifier">V</span></code> is <span class="bold"><strong>representable</strong></span>
        (can be represented) in the type <code class="computeroutput"><span class="identifier">T</span></code>,
        or, equivalently, its representation in the type <code class="computeroutput"><span class="identifier">T</span></code>
        is <span class="bold"><strong>in range</strong></span>, or <span class="bold"><strong>does
        not overflow</strong></span>.
      </p>
<p>
        Notice that a numeric type, such as a C++ unsigned type, can define that
        any <code class="computeroutput"><span class="identifier">V</span></code> does not overflow by
        always representing not <code class="computeroutput"><span class="identifier">V</span></code>
        itself but the abstract value <code class="computeroutput"><span class="identifier">U</span>
        <span class="special">=</span> <span class="special">[</span> <span class="identifier">V</span> <span class="special">%</span> <span class="special">(</span><span class="identifier">abt</span><span class="special">(</span><span class="identifier">h</span><span class="special">)+</span><span class="number">1</span><span class="special">)</span>
        <span class="special">]</span></code>, which is always in range.
      </p>
<p>
        Given an abstract value <code class="computeroutput"><span class="identifier">V</span></code>
        represented in the type <code class="computeroutput"><span class="identifier">T</span></code>
        as <code class="computeroutput"><span class="identifier">v</span></code>, the <span class="bold"><strong>roundoff</strong></span>
        error of the representation is the abstract difference: <code class="computeroutput"><span class="special">(</span><span class="identifier">abt</span><span class="special">(</span><span class="identifier">v</span><span class="special">)-</span><span class="identifier">V</span><span class="special">)</span></code>.
      </p>
<p>
        Notice that a representation is an <span class="emphasis"><em>operation</em></span>, hence,
        the roundoff error corresponds to the representation operation and not to
        the numeric value itself (i.e. numeric values do not have any error themselves)
      </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
            If the roundoff is 0, the representation is <span class="bold"><strong>exact</strong></span>,
            and <code class="computeroutput"><span class="identifier">V</span></code> is exactly representable
            in the type <code class="computeroutput"><span class="identifier">T</span></code>.
          </li>
<li class="listitem">
            If the roundoff is not 0, the representation is <span class="bold"><strong>inexact</strong></span>,
            and <code class="computeroutput"><span class="identifier">V</span></code> is inexactly representable
            in the type <code class="computeroutput"><span class="identifier">T</span></code>.
          </li>
</ul></div>
<p>
        If a representation <code class="computeroutput"><span class="identifier">v</span></code> in
        a type <code class="computeroutput"><span class="identifier">T</span></code> -either exact or
        inexact-, is any of the adjacents of <code class="computeroutput"><span class="identifier">V</span></code>
        in that type, that is, if <code class="computeroutput"><span class="identifier">v</span><span class="special">==</span><span class="identifier">prev</span></code>
        or <code class="computeroutput"><span class="identifier">v</span><span class="special">==</span><span class="identifier">next</span></code>, the representation is faithfully
        rounded. If the choice between <code class="computeroutput"><span class="identifier">prev</span></code>
        and <code class="computeroutput"><span class="identifier">next</span></code> matches a given
        <span class="bold"><strong>rounding direction</strong></span>, it is <span class="bold"><strong>correctly
        rounded</strong></span>.
      </p>
<p>
        All exact representations are correctly rounded, but not all inexact representations
        are. In particular, C++ requires numeric conversions (described below) and
        the result of arithmetic operations (not covered by this document) to be
        correctly rounded, but batch operations propagate roundoff, thus final results
        are usually incorrectly rounded, that is, the numeric value <code class="computeroutput"><span class="identifier">r</span></code> which is the computed result is neither
        of the adjacents of the abstract value <code class="computeroutput"><span class="identifier">R</span></code>
        which is the theoretical result.
      </p>
<p>
        Because a correctly rounded representation is always one of adjacents of
        the abstract value being represented, the roundoff is guaranteed to be at
        most 1ulp.
      </p>
<p>
        The following examples summarize the given definitions. Consider:
      </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
            A numeric type <code class="computeroutput"><span class="identifier">Int</span></code> representing
            integer numbers with a <span class="emphasis"><em>numeric set</em></span>: <code class="computeroutput"><span class="special">{-</span><span class="number">2</span><span class="special">,-</span><span class="number">1</span><span class="special">,</span><span class="number">0</span><span class="special">,</span><span class="number">1</span><span class="special">,</span><span class="number">2</span><span class="special">}</span></code> and
            <span class="emphasis"><em>range</em></span>: <code class="computeroutput"><span class="special">[-</span><span class="number">2</span><span class="special">,</span><span class="number">2</span><span class="special">]</span></code>
          </li>
<li class="listitem">
            A numeric type <code class="computeroutput"><span class="identifier">Cardinal</span></code>
            representing integer numbers with a <span class="emphasis"><em>numeric set</em></span>:
            <code class="computeroutput"><span class="special">{</span><span class="number">0</span><span class="special">,</span><span class="number">1</span><span class="special">,</span><span class="number">2</span><span class="special">,</span><span class="number">3</span><span class="special">,</span><span class="number">4</span><span class="special">,</span><span class="number">5</span><span class="special">,</span><span class="number">6</span><span class="special">,</span><span class="number">7</span><span class="special">,</span><span class="number">8</span><span class="special">,</span><span class="number">9</span><span class="special">}</span></code> and <span class="emphasis"><em>range</em></span>: <code class="computeroutput"><span class="special">[</span><span class="number">0</span><span class="special">,</span><span class="number">9</span><span class="special">]</span></code> (no
            modulo-arithmetic here)
          </li>
<li class="listitem">
            A numeric type <code class="computeroutput"><span class="identifier">Real</span></code> representing
            real numbers with a <span class="emphasis"><em>numeric set</em></span>: <code class="computeroutput"><span class="special">{-</span><span class="number">2.0</span><span class="special">,-</span><span class="number">1.5</span><span class="special">,-</span><span class="number">1.0</span><span class="special">,-</span><span class="number">0.5</span><span class="special">,-</span><span class="number">0.0</span><span class="special">,+</span><span class="number">0.0</span><span class="special">,+</span><span class="number">0.5</span><span class="special">,+</span><span class="number">1.0</span><span class="special">,+</span><span class="number">1.5</span><span class="special">,+</span><span class="number">2.0</span><span class="special">}</span></code> and
            <span class="emphasis"><em>range</em></span>: <code class="computeroutput"><span class="special">[-</span><span class="number">2.0</span><span class="special">,+</span><span class="number">2.0</span><span class="special">]</span></code>
          </li>
<li class="listitem">
            A numeric type <code class="computeroutput"><span class="identifier">Whole</span></code>
            representing real numbers with a <span class="emphasis"><em>numeric set</em></span>: <code class="computeroutput"><span class="special">{-</span><span class="number">2.0</span><span class="special">,-</span><span class="number">1.0</span><span class="special">,</span><span class="number">0.0</span><span class="special">,+</span><span class="number">1.0</span><span class="special">,+</span><span class="number">2.0</span><span class="special">}</span></code> and
            <span class="emphasis"><em>range</em></span>: <code class="computeroutput"><span class="special">[-</span><span class="number">2.0</span><span class="special">,+</span><span class="number">2.0</span><span class="special">]</span></code>
          </li>
</ul></div>
<p>
        First, notice that the types <code class="computeroutput"><span class="identifier">Real</span></code>
        and <code class="computeroutput"><span class="identifier">Whole</span></code> both represent
        real numbers, have the same range, but different precision.
      </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
            The integer number <code class="computeroutput"><span class="number">1</span></code> (an
            abstract value) can be exactly represented in any of these types.
          </li>
<li class="listitem">
            The integer number <code class="computeroutput"><span class="special">-</span><span class="number">1</span></code>
            can be exactly represented in <code class="computeroutput"><span class="identifier">Int</span></code>,
            <code class="computeroutput"><span class="identifier">Real</span></code> and <code class="computeroutput"><span class="identifier">Whole</span></code>, but cannot be represented in
            <code class="computeroutput"><span class="identifier">Cardinal</span></code>, yielding negative
            overflow.
          </li>
<li class="listitem">
            The real number <code class="computeroutput"><span class="number">1.5</span></code> can be
            exactly represented in <code class="computeroutput"><span class="identifier">Real</span></code>,
            and inexactly represented in the other types.
          </li>
<li class="listitem">
            If <code class="computeroutput"><span class="number">1.5</span></code> is represented as
            either <code class="computeroutput"><span class="number">1</span></code> or <code class="computeroutput"><span class="number">2</span></code> in any of the types (except <code class="computeroutput"><span class="identifier">Real</span></code>), the representation is correctly
            rounded.
          </li>
<li class="listitem">
            If <code class="computeroutput"><span class="number">0.5</span></code> is represented as
            <code class="computeroutput"><span class="special">+</span><span class="number">1.5</span></code>
            in the type <code class="computeroutput"><span class="identifier">Real</span></code>, it
            is incorrectly rounded.
          </li>
<li class="listitem">
            <code class="computeroutput"><span class="special">(-</span><span class="number">2.0</span><span class="special">,-</span><span class="number">1.5</span><span class="special">)</span></code>
            are the <code class="computeroutput"><span class="identifier">Real</span></code> adjacents
            of any real number in the interval <code class="computeroutput"><span class="special">[-</span><span class="number">2.0</span><span class="special">,-</span><span class="number">1.5</span><span class="special">]</span></code>, yet there are no <code class="computeroutput"><span class="identifier">Real</span></code>
            adjacents for <code class="computeroutput"><span class="identifier">x</span> <span class="special">&lt;</span>
            <span class="special">-</span><span class="number">2.0</span></code>,
            nor for <code class="computeroutput"><span class="identifier">x</span> <span class="special">&gt;</span>
            <span class="special">+</span><span class="number">2.0</span></code>.
          </li>
</ul></div>
</div>
<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="boost_numericconversion.definitions.standard__numeric__conversions"></a><a class="link" href="definitions.html#boost_numericconversion.definitions.standard__numeric__conversions" title="Standard (numeric) Conversions">Standard
      (numeric) Conversions</a>
</h3></div></div></div>
<p>
        The C++ language defines <span class="underline">Standard Conversions</span>
        (§4) some of which are conversions between arithmetic types.
      </p>
<p>
        These are <span class="underline">Integral promotions</span> (§4.5),
        <span class="underline">Integral conversions</span> (§4.7), <span class="underline">Floating point promotions</span> (§4.6), <span class="underline">Floating point conversions</span> (§4.8) and <span class="underline">Floating-integral conversions</span> (§4.9).
      </p>
<p>
        In the sequel, integral and floating point promotions are called <span class="bold"><strong>arithmetic promotions</strong></span>, and these plus integral, floating-point
        and floating-integral conversions are called <span class="bold"><strong>arithmetic
        conversions</strong></span> (i.e, promotions are conversions).
      </p>
<p>
        Promotions, both Integral and Floating point, are <span class="emphasis"><em>value-preserving</em></span>,
        which means that the typed value is not changed with the conversion.
      </p>
<p>
        In the sequel, consider a source typed value <code class="computeroutput"><span class="identifier">s</span></code>
        of type <code class="computeroutput"><span class="identifier">S</span></code>, the source abstract
        value <code class="computeroutput"><span class="identifier">N</span><span class="special">=</span><span class="identifier">abt</span><span class="special">(</span><span class="identifier">s</span><span class="special">)</span></code>, a destination type <code class="computeroutput"><span class="identifier">T</span></code>;
        and whenever possible, a result typed value <code class="computeroutput"><span class="identifier">t</span></code>
        of type <code class="computeroutput"><span class="identifier">T</span></code>.
      </p>
<p>
        Integer to integer conversions are always defined:
      </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
            If <code class="computeroutput"><span class="identifier">T</span></code> is unsigned, the
            abstract value which is effectively represented is not <code class="computeroutput"><span class="identifier">N</span></code> but <code class="computeroutput"><span class="identifier">M</span><span class="special">=[</span> <span class="identifier">N</span> <span class="special">%</span> <span class="special">(</span> <span class="identifier">abt</span><span class="special">(</span><span class="identifier">h</span><span class="special">)</span> <span class="special">+</span> <span class="number">1</span>
            <span class="special">)</span> <span class="special">]</span></code>,
            where <code class="computeroutput"><span class="identifier">h</span></code> is the highest
            unsigned typed value of type <code class="computeroutput"><span class="identifier">T</span></code>.
          </li>
<li class="listitem">
            If <code class="computeroutput"><span class="identifier">T</span></code> is signed and <code class="computeroutput"><span class="identifier">N</span></code> is not directly representable, the
            result <code class="computeroutput"><span class="identifier">t</span></code> is <span class="underline">implementation-defined</span>, which means that
            the C++ implementation is required to produce a value <code class="computeroutput"><span class="identifier">t</span></code>
            even if it is totally unrelated to <code class="computeroutput"><span class="identifier">s</span></code>.
          </li>
</ul></div>
<p>
        Floating to Floating conversions are defined only if <code class="computeroutput"><span class="identifier">N</span></code>
        is representable; if it is not, the conversion has <span class="underline">undefined
        behavior</span>.
      </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
            If <code class="computeroutput"><span class="identifier">N</span></code> is exactly representable,
            <code class="computeroutput"><span class="identifier">t</span></code> is required to be the
            exact representation.
          </li>
<li class="listitem">
            If <code class="computeroutput"><span class="identifier">N</span></code> is inexactly representable,
            <code class="computeroutput"><span class="identifier">t</span></code> is required to be one
            of the two adjacents, with an implementation-defined choice of rounding
            direction; that is, the conversion is required to be correctly rounded.
          </li>
</ul></div>
<p>
        Floating to Integer conversions represent not <code class="computeroutput"><span class="identifier">N</span></code>
        but <code class="computeroutput"><span class="identifier">M</span><span class="special">=</span><span class="identifier">trunc</span><span class="special">(</span><span class="identifier">N</span><span class="special">)</span></code>, were
        <code class="computeroutput"><span class="identifier">trunc</span><span class="special">()</span></code>
        is to truncate: i.e. to remove the fractional part, if any.
      </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
            If <code class="computeroutput"><span class="identifier">M</span></code> is not representable
            in <code class="computeroutput"><span class="identifier">T</span></code>, the conversion
            has <span class="underline">undefined behavior</span> (unless
            <code class="computeroutput"><span class="identifier">T</span></code> is <code class="computeroutput"><span class="keyword">bool</span></code>,
            see §4.12).
          </li></ul></div>
<p>
        Integer to Floating conversions are always defined.
      </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
            If <code class="computeroutput"><span class="identifier">N</span></code> is exactly representable,
            <code class="computeroutput"><span class="identifier">t</span></code> is required to be the
            exact representation.
          </li>
<li class="listitem">
            If <code class="computeroutput"><span class="identifier">N</span></code> is inexactly representable,
            <code class="computeroutput"><span class="identifier">t</span></code> is required to be one
            of the two adjacents, with an implementation-defined choice of rounding
            direction; that is, the conversion is required to be correctly rounded.
          </li>
</ul></div>
</div>
<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="boost_numericconversion.definitions.subranged_conversion_direction__subtype_and_supertype"></a><a class="link" href="definitions.html#boost_numericconversion.definitions.subranged_conversion_direction__subtype_and_supertype" title="Subranged Conversion Direction, Subtype and Supertype">Subranged
      Conversion Direction, Subtype and Supertype</a>
</h3></div></div></div>
<p>
        Given a source type <code class="computeroutput"><span class="identifier">S</span></code> and
        a destination type <code class="computeroutput"><span class="identifier">T</span></code>, there
        is a <span class="bold"><strong>conversion direction</strong></span> denoted: <code class="computeroutput"><span class="identifier">S</span><span class="special">-&gt;</span><span class="identifier">T</span></code>.
      </p>
<p>
        For any two ranges the following <span class="emphasis"><em>range relation</em></span> can
        be defined: A range <code class="computeroutput"><span class="identifier">X</span></code> can
        be <span class="emphasis"><em>entirely contained</em></span> in a range <code class="computeroutput"><span class="identifier">Y</span></code>,
        in which case it is said that <code class="computeroutput"><span class="identifier">X</span></code>
        is enclosed by <code class="computeroutput"><span class="identifier">Y</span></code>.
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="bold"><strong>Formally:</strong></span> <code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="identifier">S</span><span class="special">)</span></code>
          is enclosed by <code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="identifier">T</span><span class="special">)</span></code> iif
          <code class="computeroutput"><span class="special">(</span><span class="identifier">R</span><span class="special">(</span><span class="identifier">S</span><span class="special">)</span>
          <span class="identifier">intersection</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">T</span><span class="special">))</span>
          <span class="special">==</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">S</span><span class="special">)</span></code>.
        </p></blockquote></div>
<p>
        If the source type range, <code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="identifier">S</span><span class="special">)</span></code>,
        is not enclosed in the target type range, <code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="identifier">T</span><span class="special">)</span></code>;
        that is, if <code class="computeroutput"><span class="special">(</span><span class="identifier">R</span><span class="special">(</span><span class="identifier">S</span><span class="special">)</span>
        <span class="special">&amp;</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">T</span><span class="special">))</span>
        <span class="special">!=</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">S</span><span class="special">)</span></code>,
        the conversion direction is said to be <span class="bold"><strong>subranged</strong></span>,
        which means that <code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="identifier">S</span><span class="special">)</span></code> is not
        entirely contained in <code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="identifier">T</span><span class="special">)</span></code> and
        therefore there is some portion of the source range which falls outside the
        target range. In other words, if a conversion direction <code class="computeroutput"><span class="identifier">S</span><span class="special">-&gt;</span><span class="identifier">T</span></code>
        is subranged, there are values in <code class="computeroutput"><span class="identifier">S</span></code>
        which cannot be represented in <code class="computeroutput"><span class="identifier">T</span></code>
        because they are out of range. Notice that for <code class="computeroutput"><span class="identifier">S</span><span class="special">-&gt;</span><span class="identifier">T</span></code>,
        the adjective subranged applies to <code class="computeroutput"><span class="identifier">T</span></code>.
      </p>
<p>
        Examples:
      </p>
<p>
        Given the following numeric types all representing real numbers:
      </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
            <code class="computeroutput"><span class="identifier">X</span></code> with numeric set <code class="computeroutput"><span class="special">{-</span><span class="number">2.0</span><span class="special">,-</span><span class="number">1.0</span><span class="special">,</span><span class="number">0.0</span><span class="special">,+</span><span class="number">1.0</span><span class="special">,+</span><span class="number">2.0</span><span class="special">}</span></code> and
            range <code class="computeroutput"><span class="special">[-</span><span class="number">2.0</span><span class="special">,+</span><span class="number">2.0</span><span class="special">]</span></code>
          </li>
<li class="listitem">
            <code class="computeroutput"><span class="identifier">Y</span></code> with numeric set <code class="computeroutput"><span class="special">{-</span><span class="number">2.0</span><span class="special">,-</span><span class="number">1.5</span><span class="special">,-</span><span class="number">1.0</span><span class="special">,-</span><span class="number">0.5</span><span class="special">,</span><span class="number">0.0</span><span class="special">,+</span><span class="number">0.5</span><span class="special">,+</span><span class="number">1.0</span><span class="special">,+</span><span class="number">1.5</span><span class="special">,+</span><span class="number">2.0</span><span class="special">}</span></code> and range <code class="computeroutput"><span class="special">[-</span><span class="number">2.0</span><span class="special">,+</span><span class="number">2.0</span><span class="special">]</span></code>
          </li>
<li class="listitem">
            <code class="computeroutput"><span class="identifier">Z</span></code> with numeric set <code class="computeroutput"><span class="special">{-</span><span class="number">1.0</span><span class="special">,</span><span class="number">0.0</span><span class="special">,+</span><span class="number">1.0</span><span class="special">}</span></code> and range <code class="computeroutput"><span class="special">[-</span><span class="number">1.0</span><span class="special">,+</span><span class="number">1.0</span><span class="special">]</span></code>
          </li>
</ul></div>
<p>
        For:
      </p>
<div class="variablelist">
<p class="title"><b></b></p>
<dl class="variablelist">
<dt><span class="term">(a) X-&gt;Y:</span></dt>
<dd><p>
              <code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="identifier">X</span><span class="special">)</span> <span class="special">&amp;</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">Y</span><span class="special">)</span> <span class="special">==</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">X</span><span class="special">)</span></code>,
              then <code class="computeroutput"><span class="identifier">X</span><span class="special">-&gt;</span><span class="identifier">Y</span></code> is not subranged. Thus, all values
              of type <code class="computeroutput"><span class="identifier">X</span></code> are representable
              in the type <code class="computeroutput"><span class="identifier">Y</span></code>.
            </p></dd>
<dt><span class="term">(b) Y-&gt;X:</span></dt>
<dd><p>
              <code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="identifier">Y</span><span class="special">)</span> <span class="special">&amp;</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">X</span><span class="special">)</span> <span class="special">==</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">Y</span><span class="special">)</span></code>,
              then <code class="computeroutput"><span class="identifier">Y</span><span class="special">-&gt;</span><span class="identifier">X</span></code> is not subranged. Thus, all values
              of type <code class="computeroutput"><span class="identifier">Y</span></code> are representable
              in the type <code class="computeroutput"><span class="identifier">X</span></code>, but
              in this case, some values are <span class="emphasis"><em>inexactly</em></span> representable
              (all the halves). (note: it is to permit this case that a range is
              an interval of abstract values and not an interval of typed values)
            </p></dd>
<dt><span class="term">(b) X-&gt;Z:</span></dt>
<dd><p>
              <code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="identifier">X</span><span class="special">)</span> <span class="special">&amp;</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">Z</span><span class="special">)</span> <span class="special">!=</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">X</span><span class="special">)</span></code>,
              then <code class="computeroutput"><span class="identifier">X</span><span class="special">-&gt;</span><span class="identifier">Z</span></code> is subranged. Thus, some values
              of type <code class="computeroutput"><span class="identifier">X</span></code> are not representable
              in the type <code class="computeroutput"><span class="identifier">Z</span></code>, they
              fall out of range <code class="computeroutput"><span class="special">(-</span><span class="number">2.0</span>
              <span class="keyword">and</span> <span class="special">+</span><span class="number">2.0</span><span class="special">)</span></code>.
            </p></dd>
</dl>
</div>
<p>
        It is possible that <code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="identifier">S</span><span class="special">)</span></code> is not
        enclosed by <code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="identifier">T</span><span class="special">)</span></code>, while
        neither is <code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="identifier">T</span><span class="special">)</span></code> enclosed
        by <code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="identifier">S</span><span class="special">)</span></code>; for
        example, <code class="computeroutput"><span class="identifier">UNSIG</span><span class="special">=[</span><span class="number">0</span><span class="special">,</span><span class="number">255</span><span class="special">]</span></code> is not enclosed by <code class="computeroutput"><span class="identifier">SIG</span><span class="special">=[-</span><span class="number">128</span><span class="special">,</span><span class="number">127</span><span class="special">]</span></code>; neither
        is <code class="computeroutput"><span class="identifier">SIG</span></code> enclosed by <code class="computeroutput"><span class="identifier">UNSIG</span></code>. This implies that is possible that
        a conversion direction is subranged both ways. This occurs when a mixture
        of signed/unsigned types are involved and indicates that in both directions
        there are values which can fall out of range.
      </p>
<p>
        Given the range relation (subranged or not) of a conversion direction <code class="computeroutput"><span class="identifier">S</span><span class="special">-&gt;</span><span class="identifier">T</span></code>, it is possible to classify <code class="computeroutput"><span class="identifier">S</span></code> and <code class="computeroutput"><span class="identifier">T</span></code>
        as <span class="bold"><strong>supertype</strong></span> and <span class="bold"><strong>subtype</strong></span>:
        If the conversion is subranged, which means that <code class="computeroutput"><span class="identifier">T</span></code>
        cannot represent all possible values of type <code class="computeroutput"><span class="identifier">S</span></code>,
        <code class="computeroutput"><span class="identifier">S</span></code> is the supertype and <code class="computeroutput"><span class="identifier">T</span></code> the subtype; otherwise, <code class="computeroutput"><span class="identifier">T</span></code> is the supertype and <code class="computeroutput"><span class="identifier">S</span></code>
        the subtype.
      </p>
<p>
        For example:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="keyword">float</span><span class="special">)=[-</span><span class="identifier">FLT_MAX</span><span class="special">,</span><span class="identifier">FLT_MAX</span><span class="special">]</span></code>
          and <code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="keyword">double</span><span class="special">)=[-</span><span class="identifier">DBL_MAX</span><span class="special">,</span><span class="identifier">DBL_MAX</span><span class="special">]</span></code>
        </p></blockquote></div>
<p>
        If <code class="computeroutput"><span class="identifier">FLT_MAX</span> <span class="special">&lt;</span>
        <span class="identifier">DBL_MAX</span></code>:
      </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
            <code class="computeroutput"><span class="keyword">double</span><span class="special">-&gt;</span><span class="keyword">float</span></code> is subranged and <code class="computeroutput"><span class="identifier">supertype</span><span class="special">=</span><span class="keyword">double</span></code>,
            <code class="computeroutput"><span class="identifier">subtype</span><span class="special">=</span><span class="keyword">float</span></code>.
          </li>
<li class="listitem">
            <code class="computeroutput"><span class="keyword">float</span><span class="special">-&gt;</span><span class="keyword">double</span></code> is not subranged and <code class="computeroutput"><span class="identifier">supertype</span><span class="special">=</span><span class="keyword">double</span></code>, <code class="computeroutput"><span class="identifier">subtype</span><span class="special">=</span><span class="keyword">float</span></code>.
          </li>
</ul></div>
<p>
        Notice that while <code class="computeroutput"><span class="keyword">double</span><span class="special">-&gt;</span><span class="keyword">float</span></code> is subranged, <code class="computeroutput"><span class="keyword">float</span><span class="special">-&gt;</span><span class="keyword">double</span></code>
        is not, which yields the same supertype,subtype for both directions.
      </p>
<p>
        Now consider:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="keyword">int</span><span class="special">)=[</span><span class="identifier">INT_MIN</span><span class="special">,</span><span class="identifier">INT_MAX</span><span class="special">]</span></code> and <code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="keyword">int</span><span class="special">)=[</span><span class="number">0</span><span class="special">,</span><span class="identifier">UINT_MAX</span><span class="special">]</span></code>
        </p></blockquote></div>
<p>
        A C++ implementation is required to have <code class="computeroutput"><span class="identifier">UINT_MAX</span>
        <span class="special">&gt;</span> <span class="identifier">INT_MAX</span></code>
        (§3.9/3), so:
      </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
            'int-&gt;unsigned' is subranged (negative values fall out of range) and
            <code class="computeroutput"><span class="identifier">supertype</span><span class="special">=</span><span class="keyword">int</span></code>, <code class="computeroutput"><span class="identifier">subtype</span><span class="special">=</span><span class="keyword">unsigned</span></code>.
          </li>
<li class="listitem">
            'unsigned-&gt;int' is <span class="emphasis"><em>also</em></span> subranged (high positive
            values fall out of range) and <code class="computeroutput"><span class="identifier">supertype</span><span class="special">=</span><span class="keyword">unsigned</span></code>,
            <code class="computeroutput"><span class="identifier">subtype</span><span class="special">=</span><span class="keyword">int</span></code>.
          </li>
</ul></div>
<p>
        In this case, the conversion is subranged in both directions and the supertype,subtype
        pairs are not invariant (under inversion of direction). This indicates that
        none of the types can represent all the values of the other.
      </p>
<p>
        When the supertype is the same for both <code class="computeroutput"><span class="identifier">S</span><span class="special">-&gt;</span><span class="identifier">T</span></code>
        and <code class="computeroutput"><span class="identifier">T</span><span class="special">-&gt;</span><span class="identifier">S</span></code>, it is effectively indicating a type
        which can represent all the values of the subtype. Consequently, if a conversion
        <code class="computeroutput"><span class="identifier">X</span><span class="special">-&gt;</span><span class="identifier">Y</span></code> is not subranged, but the opposite <code class="computeroutput"><span class="special">(</span><span class="identifier">Y</span><span class="special">-&gt;</span><span class="identifier">X</span><span class="special">)</span></code> is,
        so that the supertype is always <code class="computeroutput"><span class="identifier">Y</span></code>,
        it is said that the direction <code class="computeroutput"><span class="identifier">X</span><span class="special">-&gt;</span><span class="identifier">Y</span></code>
        is <span class="bold"><strong>correctly rounded value preserving</strong></span>, meaning
        that all such conversions are guaranteed to produce results in range and
        correctly rounded (even if inexact). For example, all integer to floating
        conversions are correctly rounded value preserving.
      </p>
</div>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright © 2004-2007 Fernando
      Luis Cacciola Carballal<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
      </p>
</div></td>
</tr></table>
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